Quantum entanglement is nearly ubiquitous in equilibrium and non-equilibrium many-body states. Although it has been largely studied through the von Neumann entropy of a subsystem, which quantifies the entanglement between two complementary parts of a many-body system, this is not necessarily the only way. Here we review how some other measures can be fruitful in characterizing the entanglement content of many-body states. For example, we can look at the entangement between two individual spins through the concurrence or between two non-complementary, but in principle large, parts of a many-body system through the negativity. Alternatively, a quantity inspired through entanglement studies, but not itself a measure of entanglement, namely the Schmidt gap, can be effective as an order parameter for phase transitions in which only the entanglement structure of a many-body system changes. We exemplify using equilibrium states of short-range and impurity models and their quantum phase transitions.
